Electronic Communications in Probability

Block size in Geometric($p$)-biased permutations

Irina Cristali, Vinit Ranjan, Jake Steinberg, Erin Beckman, Rick Durrett, Matthew Junge, and James Nolen

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Fix a probability distribution $\mathbf p = (p_1, p_2, \ldots )$ on the positive integers. The first block in a $\mathbf p$-biased permutation can be visualized in terms of raindrops that land at each positive integer $j$ with probability $p_j$. It is the first point $K$ so that all sites in $[1,K]$ are wet and all sites in $(K,\infty )$ are dry. For the geometric distribution $p_j= p(1-p)^{j-1}$ we show that $p \log K$ converges in probability to an explicit constant as $p$ tends to 0. Additionally, we prove that if $\mathbf p$ has a stretch exponential distribution, then $K$ is infinite with positive probability.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 80, 10 pp.

Received: 14 August 2018
Accepted: 11 October 2018
First available in Project Euclid: 24 October 2018

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Zentralblatt MATH identifier

Primary: 05A05: Permutations, words, matrices 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K05: Renewal theory

regenerative permutations Bernoulli sieve

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Cristali, Irina; Ranjan, Vinit; Steinberg, Jake; Beckman, Erin; Durrett, Rick; Junge, Matthew; Nolen, James. Block size in Geometric($p$)-biased permutations. Electron. Commun. Probab. 23 (2018), paper no. 80, 10 pp. doi:10.1214/18-ECP182. https://projecteuclid.org/euclid.ecp/1540346604

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